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Section 6.3 : Volume With Rings

For problems 1 - 16 use the method disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis.

1. Rotate the region bounded by \(y = 2{x^2}\), \(y = 8\) and the \(y\)-axis about the \(y\)-axis.
2. Rotate the region bounded by \(y = 2{x^2}\), \(y = 8\) and the \(y\)-axis about the \(x\)-axis.
3. Rotate the region bounded by \(y = 2{x^2}\), \(x = 2\) and the \(x\)-axis about the \(x\)-axis.
4. Rotate the region bounded by \(y = 2{x^2}\), \(x = 2\) and the \(x\)-axis about the \(y\)-axis.
5. Rotate the region bounded by \(x = {y^3}\), \(x = 8\) and the \(x\)-axis about the \(x\)-axis.
6. Rotate the region bounded by \(x = {y^3}\), \(x = 8\) and the \(x\)-axis about the \(y\)-axis.
7. Rotate the region bounded by \(x = {y^3}\), \(y = 2\) and the \(y\)-axis about the \(x\)-axis.
8. Rotate the region bounded by \(x = {y^3}\), \(y = 2\) and the \(y\)-axis about the \(y\)-axis.
9. Rotate the region bounded by \(y = \frac{1}{{{x^2}}}\), \(y = 9\), \(x = - 2\), \(\displaystyle x = - \frac{1}{3}\) about the \(y\)-axis.
10. Rotate the region bounded by \(y = \frac{1}{{{x^2}}}\), \(y = 9\), \(x = - 2\), \(\displaystyle x = - \frac{1}{3}\) about the \(x\)-axis.
11. Rotate the region bounded by \(y = 4 + 3{{\bf{e}}^{ - x}}\), \(y = 2\), \(\displaystyle x = \frac{1}{2}\) and \(x = 3\) about the \(x\)-axis.
12. Rotate the region bounded by \(x = 5 - {y^2}\) and \(x = 4\) about the \(y\)-axis.
13. Rotate the region bounded by \(y = 6 - 2x\), \(y = 3 + x\) and \(x = 3\) about the \(x\)-axis.
14. Rotate the region bounded by \(y = 6 - 2x\), \(y = 3 + x\) and \(y = 6\) about the \(y\)-axis.
15. Rotate the region bounded by \(y = {x^2} - 2x + 4\) and \(y = x + 14\) about the \(x\)-axis.
16. Rotate the region bounded by \(x = {\left( {y - 3} \right)^2}\) and \(x = 16\) about the \(y\)-axis.
17. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(y = 2{x^2}\), \(y = 8\) and the \(y\)-axis about the
    1. line \(x = 3\)
    2. line \(x = - 2\)
    1. line \(y = 11\)
    2. line \(y = - 4\)
18. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(x = {y^2} - 6y + 9\) and \(x = - {y^2} + 6y - 1\) about the
    1. line \(x = 10\)
    2. line \(x = -3\)
19. Use the method of disks/rings to determine the volume of the solid obtained by rotating the triangle with vertices \(\left( {3,2} \right)\), \(\left( {7,2} \right)\) and \(\left( {7,14} \right)\) about the
    1. line \(x = 12\)
    2. line \(x = 2\)
    3. line \(x = -1\)
    1. line \(y = 14\)
    2. line \(y = 1\)
    3. line \(y = -3\)
20. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(y = 4 + 3{{\bf{e}}^{ - x}}\), \(y = 2\), \(\displaystyle x = \frac{1}{2}\) and \(x = 3\) about the
    1. line \(y = 7\)
    2. line \(y = 1\)
    3. line \(y = -3\)
21. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(x = 3 + {y^2}\) and \(x = 2y + 11\) about the
    1. line \(x = 23\)
    2. line \(x = 2\)
    3. line \(x = -1\)
22. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(y = 5 + \sqrt x \), \(y = 5\) and \(x = 4\) about the
    1. line \(y = 8\)
    2. line \(y = 2\)
    3. line \(y = -2\)
23. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(y = 10 - 2x\), \(y = x + 1\) and \(y = 7\) about the
    1. line \(x = 8\)
    2. line \(x = 1\)
    3. line \(x = -4\)
24. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(y = - {x^2} - 2x - 5\) and \(y = 2x - 17\) about the
    1. line \(y = 3\)
    2. line \(y = -1\)
    3. line \(y = -34\)
25. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by \(x = - 2{y^2} - 3\) and \(x = - 5\) about the
    1. line \(x = 4\)
    2. line \(x = -2\)
    3. line \(x = -9\)